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Theorem ipasslem1 26553
Description: Lemma for ipassi 26563. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  U )
ip1i.9  |-  U  e.  CPreHil
OLD
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )

Proof of Theorem ipasslem1
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0cn 10903 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  CC )
2 ax-1cn 9615 . . . . . . . . . . . 12  |-  1  e.  CC
3 ip1i.9 . . . . . . . . . . . . . 14  |-  U  e.  CPreHil
OLD
43phnvi 26538 . . . . . . . . . . . . 13  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . . . . . . . . . 14  |-  X  =  ( BaseSet `  U )
6 ip1i.2 . . . . . . . . . . . . . 14  |-  G  =  ( +v `  U
)
7 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .sOLD `  U )
85, 6, 7nvdir 26333 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
k  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
94, 8mpan 684 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  X )  ->  (
( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
102, 9mp3an2 1378 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
111, 10sylan 479 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
125, 7nvsid 26329 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
134, 12mpan 684 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  (
1 S A )  =  A )
1413adantl 473 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1514oveq2d 6324 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k S A ) G ( 1 S A ) )  =  ( ( k S A ) G A ) )
1611, 15eqtrd 2505 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G A ) )
1716oveq1d 6323 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) G A ) P B ) )
18 ipasslem1.b . . . . . . . . . . . . 13  |-  B  e.  X
19 ip1i.7 . . . . . . . . . . . . . 14  |-  P  =  ( .iOLD `  U )
205, 19dipcl 26432 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
214, 18, 20mp3an13 1381 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
2221mulid2d 9679 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2322adantl 473 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1  x.  ( A P B ) )  =  ( A P B ) )
2423oveq2d 6324 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
255, 7nvscl 26328 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  k  e.  CC  /\  A  e.  X )  ->  (
k S A )  e.  X )
264, 25mp3an1 1377 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( k S A )  e.  X )
271, 26sylan 479 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( k S A )  e.  X )
285, 6, 7, 19, 3ipdiri 26552 . . . . . . . . . . 11  |-  ( ( ( k S A )  e.  X  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
2918, 28mp3an3 1379 . . . . . . . . . 10  |-  ( ( ( k S A )  e.  X  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3027, 29sylancom 680 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3124, 30eqtr4d 2508 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) G A ) P B ) )
3217, 31eqtr4d 2508 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) ) )
33 oveq1 6315 . . . . . . 7  |-  ( ( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  ->  (
( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3432, 33sylan9eq 2525 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
35 adddir 9652 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
362, 35mp3an2 1378 . . . . . . . 8  |-  ( ( k  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
371, 21, 36syl2an 485 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3837adantr 472 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3934, 38eqtr4d 2508 . . . . 5  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
4039exp31 615 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  X  ->  (
( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  -> 
( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
4140a2d 28 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  X  -> 
( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( A  e.  X  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
42 eqid 2471 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
435, 42, 19dip0l 26438 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
444, 18, 43mp2an 686 . . . 4  |-  ( (
0vec `  U ) P B )  =  0
455, 7, 42nv0 26339 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  ( 0vec `  U
) )
464, 45mpan 684 . . . . 5  |-  ( A  e.  X  ->  (
0 S A )  =  ( 0vec `  U
) )
4746oveq1d 6323 . . . 4  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( ( 0vec `  U ) P B ) )
4821mul02d 9849 . . . 4  |-  ( A  e.  X  ->  (
0  x.  ( A P B ) )  =  0 )
4944, 47, 483eqtr4a 2531 . . 3  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) )
50 oveq1 6315 . . . . . 6  |-  ( j  =  0  ->  (
j S A )  =  ( 0 S A ) )
5150oveq1d 6323 . . . . 5  |-  ( j  =  0  ->  (
( j S A ) P B )  =  ( ( 0 S A ) P B ) )
52 oveq1 6315 . . . . 5  |-  ( j  =  0  ->  (
j  x.  ( A P B ) )  =  ( 0  x.  ( A P B ) ) )
5351, 52eqeq12d 2486 . . . 4  |-  ( j  =  0  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) )
5453imbi2d 323 . . 3  |-  ( j  =  0  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) ) )
55 oveq1 6315 . . . . . 6  |-  ( j  =  k  ->  (
j S A )  =  ( k S A ) )
5655oveq1d 6323 . . . . 5  |-  ( j  =  k  ->  (
( j S A ) P B )  =  ( ( k S A ) P B ) )
57 oveq1 6315 . . . . 5  |-  ( j  =  k  ->  (
j  x.  ( A P B ) )  =  ( k  x.  ( A P B ) ) )
5856, 57eqeq12d 2486 . . . 4  |-  ( j  =  k  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
k S A ) P B )  =  ( k  x.  ( A P B ) ) ) )
5958imbi2d 323 . . 3  |-  ( j  =  k  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) ) ) )
60 oveq1 6315 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j S A )  =  ( ( k  +  1 ) S A ) )
6160oveq1d 6323 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j S A ) P B )  =  ( ( ( k  +  1 ) S A ) P B ) )
62 oveq1 6315 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  ( A P B ) )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
6361, 62eqeq12d 2486 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) )
6463imbi2d 323 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
65 oveq1 6315 . . . . . 6  |-  ( j  =  N  ->  (
j S A )  =  ( N S A ) )
6665oveq1d 6323 . . . . 5  |-  ( j  =  N  ->  (
( j S A ) P B )  =  ( ( N S A ) P B ) )
67 oveq1 6315 . . . . 5  |-  ( j  =  N  ->  (
j  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
6866, 67eqeq12d 2486 . . . 4  |-  ( j  =  N  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
6968imbi2d 323 . . 3  |-  ( j  =  N  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) ) )
7041, 49, 54, 59, 64, 69nn0indALT 11054 . 2  |-  ( N  e.  NN0  ->  ( A  e.  X  ->  (
( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
7170imp 436 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   NN0cn0 10893   NrmCVeccnv 26284   +vcpv 26285   BaseSetcba 26286   .sOLDcns 26287   0veccn0v 26288   .iOLDcdip 26417   CPreHil OLDccphlo 26534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-grpo 26000  df-gid 26001  df-ginv 26002  df-ablo 26091  df-vc 26246  df-nv 26292  df-va 26295  df-ba 26296  df-sm 26297  df-0v 26298  df-nmcv 26300  df-dip 26418  df-ph 26535
This theorem is referenced by:  ipasslem2  26554  ipasslem3  26555  ipasslem4  26556
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